The implications of neutral evolution for neutral ecology. Daniel Lawson Bioinformatics and Statistics Scotland Macaulay Institute, Aberdeen

Transcription

1 The implications of neutral evolution for neutral ecology Daniel Lawson Bioinformatics and Statistics Scotland Macaulay Institute, Aberdeen

2 How is How is diversity Diversity maintained? maintained?

3 Talk OutLine Background: The Unified Neutral theory of Biodiversity and Biogeography Implications from an evolution model Application to Ecology Application to Pine Trees

4 What is ecological neutrality? All individuals are equivalent. (with respect to their chances of having offspring in the next generation)

5 Squawk! Probability of occurrence: When Pigs Fly

6 Why consider neutrality at all? Neutrality: theory of chance events. High observed diversities needed explaining, but no general theory. Evolution and ecology are inherently linked. Need a null model minimum model that explains diversity.

7 The Neutral Model Assume all individuals are 'equal' Valid for Phenotypes without function Genotype regions not coding for protein synthesis (12% of Human DNA is variable! Redon et al. Nature. doi: /nature05329 ) Each individual has the same probability to die (pk), or give birth (pb), in a time step For simplicity, assume the total population (N) has reached equilibrium (pk = pb) Mutations (and/or colonisation) can occur, reproduction is (a)sexual

8 The neutral model Consider N individuals each labeled by Birth or Colonise species: TIMESTEP: Pick an individual (from N) and mark it to die. Pick an individual (from N) and copy it, or with probability pm, colonise with a new species. Kill the marked individual. Same as a mutation Death

9 Making the model work Common species are common at many sites - communities don t exist in isolation Exchange individuals with a metacommunity of size Jm >> N Metacommunity composition changes much more slowly that local community

10 Full ecological model Local Community 1 Metacommunity Smaller populations Large population Species composition determined by migration Evolved by mutation Composition changes in time Static composition Local Community n Local Community 2

11 Assumptions? Fixed population size All individuals are equivalent Individual life history is irrelevant There is a speciation event

12 60 40 Jm = Species Abundance species.table(a) Results of the model 0 Pm = Simulation performed using package UNTB in R Time

13 Initial results Explains Species Abundance Distributions But Species Lifetimes for abundant species in metacommunity is impossibly long! (longer than the history of earth for a common species to be replaced worldwide)

14 Fixed ecological model Consider N individuals each labeled by species: TIMESTEP: Pick an individual (from N) and mark it to die. Pick an individual (from N) and copy. Prob. pa a proportion speciate allopatrically. Kill the marked individual. Death FISSION SPECIATION Mutate proportion of population allopatrically

15 Spatial Version Random Death Local Reproduction Local Community No need for metacommunity space takes care of it!

16 How is diversityrelation Species-Area maintained? Number of species at a given scale. A=1 A=4 A=9 D=2 D=6 D=8

17 Species Area Relation Country or Continental scale Regional scale Intercontinental scale Power law for intermediate scales

18 Diversity Time Series Normalised Diversity (10000 individuals) 50 Diversity is well defined, even though common types change constantly 0 Type Richness Simpson Index Time

19 Results of extensions Explains the power law species area relation and deviations from it Space is a satisfying explanation of metacommunity Although specific species change constantly, diversity is well defined Fission solves species lifetimes problem (but what is it?)

20 Success and failure See: J Chave, Ecol. Lett Not good for birds (they move too much) Fits non-persistent fish species but not dominant species Good fits to rainforests in Camaroon, Ecuador, Panama, Peru poor on Barro Colorado Island Hard to distinguish from distributions of very specialised species in patchy terrain.

21 Success and failure See: J Chave, Ecol. Lett Equivalence of individuals questionable (only 26% of species in one Rainforest). But per-capita averages of species often show equivalence. IN SHORT: It works more often than you d expect, but not always

22 Part 2: Observations. We expect a species : To be different enough from other species. To be constant in time. An individual of a species today is comparable with an individual of that species in the past. But how different is different enough? How constant is constant? These concepts aren t in the model!

23 The Lineage Type Space Extinct Lineages 1 family 2 species 4 types Time 7 lineages?

24 Diversity measures Measured diversity depends on diversity measure: over Species Richness: DRaw = 1 Sum species i The Number of different types Simpson Diversity: Diversity measure accounting for different rarities i 1 DS = 2 pi i Proportion of species i from total population N Rao Index: Diversity measure accounting for difference between types DRau = d ij pi p j i, j Difference between types

25 Normalised Simpson Diversity (D=28.5) Number of Types (D=117) (10000 individuals) 0.0 Normalised Diversity Diversity Time Series Time

26 Normalised Simpson Diversity N Simpson Types Normalised Rao Index (10000 individuals) All species are not equal with respect to Diversity! 0.0 Normalised Diversity Diversity Time Series Time

27 Assumptions? Fixed population size All individuals are equivalent Individual life history is irrelevant There is a speciation event

28 Part 3: Relation to Ecology

29 Phenotype Distribution Consider 1 dimensional case: mutations can be either to the left or to the right. Expected pattern is a Normal Distribution: Mean Proportion of individuals But Variance Pattern is produced by Selection. Observable (Height, weight, etc)

30 Test Problem Normal Distribution? QUESTIONS: Are spots functional in Imaginarius Forma? How many types/species do we have? What action should be taken to save spotty variety? Rare Spotty Variety

31 An evolution model Consider N individuals each labeled by phenotype position: TIMESTEP: Pick an individual (from N) and mark it to die. Pick an individual (from N) and copy it. With probability pm Mutate to a similar type. Kill the marked individual.

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35 Test Problem ANSWERS: Can be neutral. - Spots might not be functional. - Only one species in any real sense. - Saving Spotty variety requires spatial segregation from Less Spotty variety.

36 Solution Simplify the model consider only first two moments of the distribution. Peak is a Gaussian distribution of area 1 with dynamic mean µ and width w. Select death location x Select birth location y, mutated by 1 with probability Pm Remove 1/N from death location and place at birth location Update µ and w

37 Solution method Write down equations for the change in the mean and the variance of the peak position µ and the width w. Take continuous limit to obtain Stochastic Differential Equations. Solve

38 Neutral Phenotype Results Width of peak is proportional to fluctuations in the width of the peak. Corresponds to multiple clusters Peak position drifts with constant speed when population size changes. Evolution speed is the same in small and large populations! Obtain an analytic solution to act as a null hypothesis. Clearly, differences between types matter!

39 Fission Speciation Consider N individuals each labeled by species: Fission ASSUMES a neutral drift process. But differences can t change without a death!? Death FISSION SPECIATION Mutate proportion of population allopatrically

40 Fix fission speciation? Barriers (move in time) Random Death Local Reproduction

41 Implications? There is no natural species definition though arbitrary cut-offs still work. (Following holds in sexual case, where there is a natural species definition) No speciation event but a speciation process. Fission speciation makes little sense in this context and the fix is complex. So: neutral ecological model is not parsimonious for the metacommunity.

42 Full ecological model Local Community 1 Metacommunity Smaller populations Not well modelled by neutral evolution! Species composition determined by migration BAD NULL MODEL Composition changes in time Local Community 2 Local Community n Local community processes are consistent with the neutral model GOOD NULL MODEL

43 Part 4: Application to Pine Trees Pine trees produce varying monoterpenes. Large diversity observed within a forest. Observed forests are remnants of much larger historical forests Metacommunity concept relevant Neutrality is a good null model within a single species. But monoterpenes can effect sapling mortality which effect is most important?

44 (not) A neutral model Work with Colin Beale and Jack Lennon Trees grow at a given location And compete for resources. Therefore future success is driven by intensity of competition. Neutral model with respect to genotype, but not individuals. Resolves the problem of non-observation of individual level equivalence.

45 Model details Trees grow and compete for space Trees produce seeds, which disperse Seeds are pollinated by other trees, whilst still on the mother tree Diversity of local forest is maintained by occasional external pollen Monoterpene production is heritable

46 Colour represents terpene concentrations similar colour similar terpenes recent ancestry

47 Log-Likelihood Max Log-likelihood Proportion of runs more likely than data Probability of seeds coming from outside

48 Conclusions Neutrality is a useful concept for null models Ecological models can be informed by evolution Speciation event - examined more closely Null models are useful to inform which processes are interesting

49 Beyond Neutrality Compare with other models Deterministic Differential Equation Models Stochastic models with selection Network models, etc. Neutrality is not for life its just for Christmas! Solves some problems but is just a null model!

50 References Hubbell: Unified neutral theory of biodiversity and biogeography, 2001 Chave: Neutral Theory and Community Ecology, Ecology Letters, (2004) 7: Lawson and Jensen: Neutral Evolution as Diffusion in phenotype space: reproduction with mutation but without selection Physics Review Letters, March 07 (98, ) bio/ Package UNTB for R Thank you for your attention!

51 Ecological Model Results Number of individuals Species number Θ = (population size)(probability of a new species) ordered by size

52 So what is diversity? Ecological Sense: number of different species or types Requires definition of species: Biological Species concept? Phenotypically distinct? Genotypic species concept? Definition of genotypic species is arbitrary: Cut-off in time to last common ancestor Need a difference based measure.

53 Species Area Relation 2 Diversity (Log Scale) Simpson Index on Species Simpson Index on Types Raw Species Count Raw Type Count Rao Index Area (Log Scale)

54 Test Problem Mean Variance

55 Time average is a Normal Distribution when measured relative to current mean position - No higher order effects, on average

56 Solving for the width Mutation distance d (w ) = 2 Change in variance (in a = * 2w p N Generation time dt 2w + dw N Deterministic part + Noise part 2 dw is Random, mean 0 timestep) Solution at steady state: 2 Npm ( Npm ) 2 w2 p( w)dw e dw 5 2w Power-law decay at large w 2

57 Neutral Clustering results Mean width: w Position: x Npm 8 T ( pm w ) < 2 RMS Fluctuations in w also ~ N0.5 With time in generations... <x>rms is independent of N! pmt 2 Compare with diffusion: x RMS pmt N w RMS pmt N Diffusion does nothing in infinite populations... evolution does more!

58 Variogram gamma Variogram gamma 0.0e e e-05 Variogram gamma tricyclene Distance a.pinene 40 Distance b.pinene 40 Distance

59 [Algebra]